A numerical study of tadpole swimming in the wake of a D-section cylinder ＊

Hao-tian Yuan (袁昊天), Wen-rong Hu (胡文蓉)

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Jiao Tong University and Chiba University International Cooperative Research Center (SJTU-CU-ICRC), Shanghai Jiao Tong University, Shanghai 200240, China MOE Key Laboratory of Hydrodynamics, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: haotian-yuan@outlook.com

Introduction

The living environment of aquatic animals is complicated and ever-changing. After generations of evolution, the aquatic animals can naturally take advantages of the surrounding environment. By adjusting the body to control the flow around, they can reduce the locomotory cost, with high swimming efficiency. A well known example is that when a cluster of schooling fish are swimming, the fish in the downstream consumes less energy[1-3]. Benefits can also be gainedby aquatic animals interacting with vortices from the fluid flow passing stationary objects.

How the fish extracts energy from environmental vortices is a topic of considerable interest. The inter- action between the fish and the vortices shed from a cylinder might provide some insight on how fish swim in complex flows. Gopalkrishnan et al.[4]and Shao and Pan[5]studied the interactions between the flapping foil and the vortices shed from the D-section cylinder. Three typical modes were found in the wake: the expanding wake, the destructive interaction and the constructive interaction.

Experimental studies of the fish swimming in the vortex wakes were carried out in recent years. Beal et al.[6]found that a dead and flexible fish behind the D-cylinder is propelled upstream when resonating with the vortices. Fish and Lauder[7]found that the vortex wakes may lead to a passive propulsion to the subject in the wake. Liao et al.[8-10]compared fish swimming in the wake of a D-cylinder to those swimming in a free stream. It was shown that a novel body kinematicsis involved for the fish behind a D-cylinder, termed the Kármán gait. The fish changes its undulating frequency and wave-length to synchro- nize the vortices shed from the D-cylinder, slaloming between the vortices shed from the D-cylinder rather than swimming through them. On the other hand, many numerical simulations for fish interacting with the environmental vortices were carried out. In these studies, undulating foils were often used as a sim- plified fish model. Dong and Lu[11]used the space- time finite element method to simulate the propulsive performance and the vortex shedding of fish-like travelling wavy-plates. Deng et al.[12,13]studied two traveling wavy foils in a tandem arrangement and the hydrodynamics in a diamond shaped fish school. Shao et al.[14]investigated the hydrodynamic performance of an undulating foil in vortex wakes of a D-cylinder. The wake area can be divided into three domains: the suction domain, the thrust enhancing domain, and the weak influence domain. Wu[15-17]numerically investi- gated the flow characteristics around a stationary cy- linder with an attached undulating plate or a detached undulating plate.

Compared to the fish with streamlined bodies, less attention was paid to a tadpole swimming. As the larva of the frogs and toads, the abrupt transition from their globose bodies to the laterally compressed tails makes them seem less “streamlined”[18]. In fact, tadpoles are good at swimming. The lateral deflec- tions at the tadpole snout helps to generate thrust[19]. Wassersug[20]measured the propulsive efficiency of a tadpole and found that the efficiency is as high as that of fish. Liu et al.[21]found that the tadpoles’ ability of efficient swimming is attributed to their high- frequency and large-amplitude undulation. However, very little is known about the interaction between the tadpole and its surroundings.

This paper numerically studies the vortex struc- ture and the hydrodynamic performance of a tadpole undulating in the wake of a D-section cylinder. In addition, the effects of various controlling parameters on the hydrodynamic performance of an actively undulating tadpole model with no forward motion are also investigated.

1. Materials and methods

1.1 Physical model

The two-dimensional simulation is conducted for the tadpole swimming. Based on the observed data[22], the tadpole model is built as shownin Fig.1. Just like previousstudies[5,14], a tadpoleis assumed to undulate actively without forwardmotion in the wake of a stationary D-section cylinder. Figure 2 shows the tadpole behind the D-cylinder. In the study, the distance between the D-cylinder and the tadpole is denotedby S. The diameter of the D-cylinder is D. The origin body length of the tadpole is L. The Reynolds number is defined as R e = U L /ν, where ν is the kinematic viscosity of water, and the reference velocity U is the velocity of the oncoming flow.

In this study , the tad pole model is simplified to beanon-physicalmodel.Thedistance in the x direc-

Fig.1 The tadpole model illustrated by Köhler et al.[22]

Fig.2 The tadpole behind a D-cylinder

tion between the head and the tail tip is keptto be L, which means the body length of the tadpole extends a little to keep the distance of L. As the vortex struc- tures and the forces on the tadpole are mostly deter- mined by the amplitude and the frequency of the undulating motion, and the extension of the tadpole is very little, this modification affectsvery little the simulation results.

The motion of the tadpole can be described by the following function

where λ is the wave-length, which is set to be 0.87L in the study according to Wassersug[20]. φ represents the phase angle. t is the time and f is the frequency of the tadpole undulation.It is noted that the undulating mode of the tadpole is much different from that of the fish. The undulatory amplitude of a tadpole is much larger than that of fish. Additionally, the head of a tadpole usually oscillates in the swimming while the fish’s head has almost no oscillation. The amplitude of the tadpole’soscillation along the length, ()A x, is calculated by the spline in- terpolation from the original data[20]listed in Table 1.

Table 1 Five maximum amplitudes along the tadpole length

1.2 Numerical method

The Reynolds number based on the tadpole body length is 2 000 in this study, which matches the swimming environment of a small tadpole. Hence the flow can be treated as a laminar flow. The finite volume method is used to solve the Navier-Stokes (N-S) equations for the unsteady incompressible viscous flow. In this study, a special dynamic mesh fitting deforming body surface at each time step is employed to match the tadpole’s high-amplitude periodic motion, as shown in Fig.3(a). The PIMPLE method, a combination of the semi-implicit method for pressure-linked equations (SIMPLE) method and the pressure implicit with splitting of operators (PISO) method, is used to solve the unsteady flow. The SIMPLE method is used to solve the N-S equations within each iteration step, while the PISO method is used for the time marching. In addition, the valida- tions of the time step and the grid independence are conducted. As a result, the efficiency and the accuracy of the simulation are well guaranteed. Figure 3(b) shows the computational domain of this study. The no-slip boundary condition is specified on the edges of the cylinder and the tadpole. On the inlet, the boundary condition is set as the velocity inlet condition. On the outlet, the boundary condition is set as the outflow condition, where ∂ U /∂ n=0 and ∂p/∂ n=0 (Urepresents the velocity and p represents the pressure). The symmetry boundary condition is set on the up and down bounds.

Fig.3 (Color online) Dynamic mesh fitting the tadpole’s de- forming body surface and the computational domain

1.3 Force coefficients

In this study, the hydrodynamic performance of the undulating tadpole in the vortex wakes is numeri- cally studied. The thrust coefficient (CT), the lift coefficient (CL) and the power coefficient (CP) at each time step are calculated.

where FTis the tadpole’s total thrust force in the opposite direction of the x -axis, FLis the tadpole’s total lift force in the y -axis direction, P is the power cost of the tadpole’s undulating swimming, and ρ is the density of the water. Furthermore, the mean thrust coefficientand the mean power coeffi- cientin one period are calculated.

2. Numerical validation

A NACA0012 airfoil undergoing a fish-like undulating motion in either a free stream (FS) or vortex wakes (VW) is simulated at Reynolds number of 1 500. The results are compared with the numerical results of Shao et al.[14]. Table 2 shows the parameters and the mean thrust coefficient indifferent cases. It is shown thatthe numerical methods employed in this study are reliable for the simulation of undulating objects.

Table 2 Parameters and results ofthe mean thrust coeffi- cient indifferent cases

3. Results and discussions

Unlike streamline bodies, the tadpole has a large amplitude and a high frequency when undulating. And this large-amplitude and high-frequency undulating motion is seem to be “customized” for the tadpole. Our results show that the vortex structure and the flow characteristics of a tadpole undulating behind a D-cylinder are different from those of fish[14]because of the differences of their body shape and the motion mode. The main differences of the motion mode arein the amplitude and the frequency of the undulating. In the studies of a fish model undulating in vortex streets, the amplitude of undulating is relatively small while the frequency of undulating is relatively low and close to the vortex shedding frequency of the cylinder. However, the tadpole model with larger amplitude and higher frequency has the ability of totally breaking the incoming vortices at the tail tip. And through the breaking, the tadpole gains a lot of thrust.

In Table 3, theresults of a tadpole and a NACA0012 airfoil undulating in the tadpole way are presented. In both cases, R e = 2 000, f L / U =0.3, S/ L = 1 and D = 0.3. As seen in the table, when the amplitude is large and the frequency is high, the tadpole model has a better thrust effect than the NACA0012 model. The results show that the motion mode of the tadpole matches its non-streamlined body shape very welland provides the tadpole with an effective thrust when undulating.

Fig.4 (Color online) Flow field of an undulating tadpole swimming in free stream (UF)

Table 3 Parameters and hydrodynamic performances of a tadpole and an airfoil undulating in the tadpole way

3.1 Comparisons ofdifferent swimming motions

In this subsection, we compare the tadpole swim- ming in different motions, such asa non-undulating tadpole gliding in the cylinder wake (GW),an undula- ting tadpole swimming either in a free stream (UF) or in vortex wakes (UW). The parameters and the hydrodynamic performance of these cases are listed in Table 4.

The Reynolds number here based on the tadpole body lengthis 2 000, and under that condition for the tadpoleswimming in nature,the undulating frequency is fL/ U =2.0. The undulating tadpole in the cylinder wake can gain the highest thrust. Besides, as the non- undulating tadpole glides in the wake of a cylinder, the tadpole is also subject to a weak thrust.

It is well known that the thrust of a tadpole swimming in a free stream is generated by a backward jet, which is formed by the reserve Kármán vortex streetin the wake (Fig.4). When a non-undulating tadpole glides in the wake of a cylinder, only weak vortices are formed behind the snout of the tadpole. There is still a Kármán vortex street in the wake. However, there are low pressure regions behind the cylinder. And the pressure in front of the tadpole is a little lower than that behind the tadpole (Fig.5). Thus, the tadpole is subject to a small thrust despite of its blunt body. When a tadpole undulates in the cylinder wake, the vortex structure in the wake is much more complex than that in other cases, which will bedis- cussed in the following sections. The undulating tadpole destroys the Kármán vortex street of the cylinder (Fig.6). As a result, the passing vortices from the D-cylinder are stronger and much closer to the tadpole body. These vortices produce a suction force forthe tadpole. Further, local reverse flows are generated by the passing vortices on both side of the tadpole surfaces, leading to a local forward friction force. Additionally, a local high-speed backward jet can be found behind the tadpole, too.

Therefore, it may be concluded that there are three main factors leading to the increase of the thrust of the tadpole behind a D-cylinder: the backward jet in the wake, the local reverse flows on both sides of the tadpole and the suction force caused by the passing vortices.

3.2 Effect of relative undulating frequency

It is found in this study, that the change of tadpole’s undulating frequency has almost no in- fluence on the frequency of the vortices shed from D-cylinder, which is represented by f0. Here f0is always equal to 0.641. Thus the Strouhal number of shed vortices from the cylinder St0= Df0/U is. 0.1923 In this subs ecti on, the effec t of the relative un- dulating frequency(f/ f0) onthehydrodynamic per- formance will be discussed.

Table 4 Parameters and hydrodynamic performance in different swimming motions

Fig.5 (Color online) Flow field of a tadpole gliding in cylinder wake (GW)

Fig.6 (Color online) Flow field of an undulating tadpole swimming in cylinder wake (UW)

Table 5 Parameters and hydrodynamic performances in the cases of different frequencies

3.2.1 The dodging mode

In the case UW2, a tadpole undulates behind the D-cylinder at the frequency exactly the same as the vortex shed frequency of the D-cylinder. In this case, the tadpole is subject to a very small drag with little power consumption (Table 5).

Fig.7 (Color online) Flow field of case UW2

Figure 7 shows the vorticity contours and the pres- sure contours of the tadpole in this case. The flow pattern is very special. In a whole period, the tail tip of the tadpole exactly dodges the vortices from the D-cylinder, rather than breaking them. In addition, the vortices formed by the tadpole undulating movement are greatly weakened. There is no vortex shed from the tadpole tail. Therefore, the wake behind the tadpole is still a Kármánvortex street, which contri- butes to the drag. Although the lower pressure regions move downstream with the passing vortices from the cylinder, the lower pressure regions are not quite close to the body surface. As a result, the undulating tadpole is subject to a drag. But why is the undulating tadpole subject to a drag while the non-undulating tadpole (GW) generates a small thrust in the vortex wakes? Maybe it is because the undulating movement itself brings a drag, but the high-speed jet stream caused by the undulating movement generates alarger thrust. Then in a case where a tadpole is undulating but not able to produce a high-speed backward jet stream, it could be subject to a drag.

Fig.8 Time histories of the thrust coefficients

On the other hand, it is noted thatin this case is the lowest among all cases, which means that the tadpole consumes very little energy to maintain the undulation. Asimilar phenomenon was found in the experiments of Liao[10], that the fish swimming in the wake of a D-cylinder adopts the Kármán gait by changing its undulating frequency and wavelength to synchronize the vortices and slaloming among them to save the energy cost. Hence, when a tadpole undulates in the wake of the D-cylinder at the vortex shed frequency of the cylinder, the tadpole may also adopt a dodging mode to save the energy cost.

3.2.2 The breaking mode

In the case UW, the undulatory frequency is the highest. The tadpole gains the highest thrust but with the hi ghest ener gy cos t (Table 5). When the undu- latingfrequencyofthetadpoleis high enough,more vortices are shed from the tail tip than from the D-cylinder. So the tadpole is capable of breaking all vortices from the D-cylinder, to create a much more complicated flow field.

In the case UW, as in Fig.6, pairs of positive and negative vortices are shed from the tail tip in each undulating period. Vortices from the D-cylinder move along the tadpole body, together with the vortices generated by the tadpole head oscillation. The lower pressure region on the tadpole head is larger than that in other lower frequency cases. As the passing vortices move from the head to the tail of the tadpole, they generate local reverse flows on both sides of the tadpole surfaces. When the vortices reach the tail tip, they are forced to deform into a narrow and long shape and stay around the tail tip waiting for a vortex created by the undulation in the same direction to merge together. In the wake, although the arrangement of vortices does not have a regular pattern, a positive vortex is always above a negative vortex, to generate a local backward jet, which can contribute to the thrust for the tadpole. When the vortices deform and merge at the tail tip, the strength of lower pressure regions created by these vortices is weakened. When the case UW is compared to other cases with low frequencies, it is shown that the high-frequency undulation gene- rates a backward jet stream in the wake. Therefore the low pressure region in front of the head, the local reverse flows on both sides of the tadpole and the high-speed backward jet stream in the wake all help the tadpole generate the most thrust, while the mean power coefficient is also the largest.

Fig.9 (Color online) Flow field of case UW3

Fig.10 (Color online) Flow field of case =0.4S L

3.2.3 The dodging mode alternatingwith the breaking mode

When the undulating frequency is not very high except in the case UW2, the characteristics of the flow field change with time to come to the dodging mode alternating with the breaking mode. Here we take the case UW3 for an example.

In the case UW3, the frequency of the tadpole undulation is slightly higher than the vortex shed fre- quency from the D-cylinder.

As seen in Fig.8,TC varies in a large range, showing complicated periodic trends. For the time from 60 to 63, the tadpole generates a high thrust. Forthe time from 70 to 73, the thrust is much lower. And sometimes the tadpole is subject to a drag.

At the moments of 61.2 and 61.5, the tadpole breaks the Kármán vortex street of the D-cylinder. As seen in Fig.9(a), vortices are shed from both the head and the tail tip of the tadpole. The vortices shed from the head are weakerthan those shed from the tail tip. The vortices from the D-cylinder move along the tad- pole body to generate lower local reverse flows on both sides of the tadpole, which contributes to the thrust. When the vortices of the D-cylinder reach the tail tip of the tadpole, they are deformed and stay around the tail tip until a vortex in the same direction is formed due to the tail tip undulation. Then these vortices in the same direction merge together, shed from the tail. Vortices in the wake of the tadpole are weak and no obvious regions of low-pressure are found in the wake of the tadpole. Thus, the pressure difference between the head and the tail tip is high.

Two main factorsmight account for the high thrust during this time period. Firstly, the pressure around the head is much lower than that around the tail tip. Secondly, the vortices on both sides of the tadpole generate local reverse flows, especially those at the tail tip, which deform and merge directly on the surface of the tadpole for a relatively long time. The local reverse flows generate friction forces in the direction opposite to the main stream. At the moments of 70.6 and 71.0, the tadpole dodges vortices from the D-cylinder. As seen in Fig.9(a), the vortices from the D-cylinder are not forced to deform at the tail tip. There are small and weak vortices shed from both the head and the tail tip of the tadpole, which fade away in the wake quickly. The wake of the tadpole is still a Kármán vortex street like the wake of the D-cylinder. In Fig.9(b), each vortex generates a low-pressure region. There are still obvious low-pressure regions in the wake of the tadpole as the vortices do not deform and merge at the tail tip. Because the tadpole dodges vortices, the pressure difference between the head and the tail tip is smaller, and the local reverse flows have less effect on the tadpole. As a result, the thrust of the tadpole is much lower.

In the case UW1, the flow field is quite similar to that of the case UW3 despite of the difference of the undulating frequency. The tadpole’s breaking mode alternates with the dodging mode. The vortex strength in the lower frequency case is lower than that in the higher frequency case. Hence compared to the case UW3,andin the case UW1?lower (Table 5).

3.2.4 Summary

When a tadpole undulates in the wake of the D- cylinder, it may adopt the breaking mode to destroy vortices or the dodging mode to slalom among vortices. The flow fields and the hydrodynamic per- formances of these two modes are quite different. In the breaking mode, the tadpole has a largermean thrust coefficient and mean power thrust coefficient, which means that the tadpole generates a larger thrust but also consumes more energy to maintain the undulating swimming. On the contrary, in the dodging mode, the tadpole is subject to a low thrustor even a little drag. However, it can maintain undulating swimming in the wake with very little energy consumption.

3.3 Effect of the distance between the D-cylinder and the tadpole

In this subsection, the undulating frequency is relatively high, which ensures that the tadpole could break each vortex shed from the D-cylinder to gener- ate the thrust. In all cases, R e = 2 000 and f L/ U=0.3. Among all cases, three typical kinds of wakes are observed. The parameters and the hydrodynamic performance of in the cases with different distances are listed in Table 6. The thrust and energy cost are the highest when the distance between the tadpole and cylinder is 0.6L

Table 6 Parameters and hydrodynamic performances in the cases with different distances

3.3.1 No vortex street in the wake of D-cylinder

When / ＜ 0.5S L , there is no vortex street in the wake of the D-cylinder. The flow field is similar to that of a tadpole in afree stream.

In Fig.10(a), the vortex street of the D-cylinder is not formed. Instead, thevortices move in the direction of the stream and is attached to the surface of the tadpole. As the tadpole undulates, pairs of positive and negative vortices are shed from the tail tip, forming a reserve Kármán vortex street. The stable high-speed backward jet stream formed by the reverse Kármán vortex street makes the main contribution to the tadpole thrust. Besides, as shown in Fig.10(b), there is a low pressure region between the D-cylinder and the tadpole.

Compared to the case in a free stream (UF), the increase in the mean thrust coefficient can have the following two explanations. The vortex sheet of the D-cylinder increases the vorticity intensity in the wake, resulting in a stronger high-speed backward jet and the subsequent higher thrust. Furthermore, the exis- tence of a low pressure region between the D-cylinder and the tadpole makes the pressure around the tail tip higher than that around the head. The pressure difference provides the suction forces.

3.3.2 Kármán vortex street between the D-cylinder and the tadpole

When / 1.0S L＞ , vortices are shed from the D-cylinder, forming a Kármán vortex street.

As seen in Fig.11(a), after shedding from the D-cylinder, the vortices expand in the -yaxis direc- tion, forming a complete Kármán vortex street. The vortices do not deform and enlarge when reaching the head of the tadpole. Vortices on the tadpole body generate local reverse flows. At the tail tip, the vortices are deformed, enlarged and merge with the vortices caused by the motion of the tail tip in the same rotating direction. At the tip of the tadpole tail, a positive vortex is above a negative one, to generate a local high-speed jet stream in the wake. In the wake, the arrangement of vortices is irregular. In Fig.11(b), each vortex generates a region of low pressure. The vortices from the D-cylinder have a larger magnitude than those from the tail tip.

Fig.11 (Color online) Flow field for the case =1.6S L

The thrust comes also from the friction forces caused by the local reverse flows, the suction forces caused by the pressure difference between the head and the tail and the local high-speed backward jet in the wake. As the distance between the D-cylinder and the tadpole increases, the vorticity magnitude decreases when they reach the tadpole. As a result, the thrust of the tadpole will decrease slightly.

3.3.3 Incomplete Kármán vortex street between the D-cylinder and the tadpole

When 0.5＜ / ＜1.0S L , vortices are shed from the D-cylinder, forming an incomplete Kármán vortex street in the wake of the cylinder.

In Fig.12 positive and negative vortices are periodically shed from the D-cylinder. The vortices reach the head of the tadpole without expanding in the y-axis direction. The vortex structures are similar to those inthe case of S =0.8L. The differences lie in the strength and the relative position of the vortices.

Fig.12 (Color online) Flow filed of case =0.6S L

When vortices form an incomplete Kármán vortex street behind the D-cylinder, the tadpole generates the most thrust among all three kinds of the wakes. There may be three explanations. Before the vortices of the D-cylinder reach the head of the tadpole, they are still near the centerline of the D-cylinder, which makes vortices closer to the tadpole than in other cases. The localreverse flows on the tadpole surface are the strongest, hence the friction forces opposite to the stream direction is of much larger magnitude. Due to the short distance between the D-cylinder and the tadpole, a low pressure region is formed with a large range and large magnitude, which provides the tadpole with large suction forces.

In a short distance, the vortices from the D-cylinder are still ofa large magnitude. As a result, the local high-speed jet stream in the wake caused by vortices is also of a large magnitude.

3.3.4 Summary

Three kinds of wakes are observed at different distances between the D-cylinder and the tadpole. As seen in Table 3, when undulating in a relative high frequency, a tadpole has larger mean thrust coefficients and mean power coefficients behind a D-cylinder than that in a free stream. As shown in Table 5,these two coefficients increase at first then decrease with the increase of the distance. When the distance is appropriate for the vortices shedding from the D-cylinder to form an incomplete Kármán vortex street,the vorticity magnitude of the vortices is the largest,resulting in the strongest local reverse flows, the highest suction force and the strongest local highspeed jet stream in the wake. All these contribute to the highest thrust. It is found that, when the distance between the D-cylinder and the tadpole is 0.6 tadpole’s length (S =0.6L), the tadpole has the largestmean thrust coefficient and mean power coefficient.

4. Conclusions

In this study, two-dimensional simulations are performed to investigate the tadpole undulating swimming in the wake of a D-sector cylinder. Thevortex structure of the flow field and the hydrodynamic performance of the tadpole arestudied.

When the tadpole undulates in the wake of the D-cylinder, three main explanations might be made for the increase of the thrust: the backward jet in the wake, the local reverse flows on the tadpole surface and the suction force caused by the pressure difference between the head and the tail tip of the tadpole.

The relative undulating frequency of the tadpole has great influences on the hydrodynamic performance. At different undulating frequencies, the tadpole breaks or dodges the vortices from the D-cylinder. When the tadpole breaks vortices, the vortices get deformed and merged with the vortices formed by the tadpole undulating. The tadpole generates a great thrust but consumes much energy to maintain undulating. On the contrary, when the tadpole dodges the vortices, the vortices formed by the tadpole undulating is greatly weakened. The tadpole is subject to a small thrust or even a drag while the energy consumption is very little. In general, the breaking mode provides a great thrust and the dodging mode saves energy for the tadpole undulating swimming.

When the tadpole is located at different distances behind the D-cylinder, three typical kinds of wakes are observed, which are classified by the vortex structures behind the D-cylinder. As the distance between the D-cylinder and the tadpole increases, the mean thrust coefficient and the power coefficient increase at first then decrease. When an incomplete Kármán vortex street forms between the D-cylinder and the tadpole, the tadpole is subject to the highest thrust.

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